The description of the transport of charge carriers in the oxide layer proposed by the DPCM is similar to the transport of charge carriers in a semiconductor device as proposed by van Roosbroeck [1]}. The differences come first from some reaction terms due to recombination-generation in the continuity equations and mainly from the boundary conditions, which are of Dirichlet-Neumann type in the semiconductor setting. The mathematical analysis of the drift-diffusion-Poisson system of equations for semiconductor devices is the subject of a series of seminal papers by Gajewski and Gröger [2]}, [3]}, [4]}. The strategy of proof proposed by Gajewski and Gröger relies strongly on the underlying variational structure of the model, in agreement with thermodynamics. One keypoint is a convex functional which can be interpreted as a free energy from the viewpoint of thermodynamics . This strategy has been further used in many papers dealing for instance with electro-reaction-diffusion processes [5]}, [6]}, [7]}, spin-polarized drift-diffusion models [8]}, electronic models for solar cells [9]}, [10]},...
